The stationary equilibrium of three-person coalitional bargaining games with random proposers: a classification

We present a classification of all stationary subgame perfect equilibria of the random proposer model for a three-person cooperative game according to the level of efficiency. The efficiency level is characterized by the number of “central” players who join all equilibrium coalitions. The existence of a central player guarantees asymptotic efficiency. The marginal contributions of players to the grand coalition play a critical role in their expected equilibrium payoffs.

1. Recently, Nash (2008) considered a non-cooperative bargaining model called the agencies method for a three-person cooperative game and presented some computational results.

2. It is also well known that an efficient allocation is guaranteed if renegotiation is allowed. See Seidmann and Winter (1998) and Okada (2000), among others.

3. This stopping rule loses no generality of analysis for our aim to study a three-person game where, if a two-person coalition forms, then one player outside the coalition has no choice except receiving the zero value. A general rule used in Okada (1996) allows sequential formation of coalitions in an \(n\)-person game.

4. The players’ responses depend surely on the proposal in the present round.

5. For simplicity of notation, we assume without loss of generality that a central player \(k\) is the same in the sequence \(\{ \sigma ^\delta \}\). If not, choose such a subsequence of it. This is possible since the number of players is finite. It does not matter for the proof whether or not \(k\) is always the same.

6. We have simplified the set notation \(\{1, 2, 3 \}\) to \(123\) in Table 1. Similar notation is used in this section.

7. The list of 162 possible configurations is available upon request.

8. The first inequality can be derived as follows. Let \(r_1\) and \(r_1'\) be the probabilities that player 1 chooses coalitions 12 and 13, respectively, let \(r_2\) and \(r_2'\) be the probabilities that player 2 chooses coalitions 12 and 23, respectively, and let \(r_3\) and \(r_3'\) be the probabilities that player 3 chooses coalitions 23 and 13, respectively. Since \(\theta _1 = 1 - (r_2' + r_3)/2\), \(\theta _2 = 1 - (r_1' + r_3')/2\), and \(\theta _3 = 1 - (r_1 + r_2)/2\), we have \(\theta _1 + \theta _2 + \theta _2 = 3 - ((r_1 + r_1') + (r_2 + r_2') + (r_3 + r_3'))/2 > 3/2\).

9. Subcases (i) and (ii) are degenerate in the sense that the coalitional values \(v(S)\) satisfy some equality constraint.

10. There is another constraint, \(\theta _2 + \theta _3 = 1\), which we omit for simplicity of exposition.

11. We can compute \(\theta _2 = (3v(12) + \delta v(13) - \delta v(23) - (9 + 3 \delta )v_1)/(2 \delta v(12) - 6 \delta v_1)\) and \(\theta _3 = (\delta v(12) + 3v(13) - \delta v(23) - (9 + 3\delta )v_1)/(2 \delta v(13) - 6 \delta v_1)\). Since it is cumbersome to derive a general formula for the expected equilibrium payoffs in subcases (iii) and (iv), we have omitted this derivation.

Authors and Affiliations

1. Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan

   Akira Okada

Corresponding author

Correspondence to Akira Okada.

I am grateful to two anonymous referees for helpful comments. I would also like to thank Takeshi Nishimura for excellent assistance with this research. Financial support from the Japan Society for the Promotion of Science under Grant No. (S)20223001 is gratefully acknowledged.

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